‘With me being one of them’—grammatically correct?

Would the phrase 'With me being one of them' be grammatically correct? Sounds a bit odd in my head and I triple-checked mentally but couldn't tell if it was correct or not.

'With I being one of them' doesn't sound correct either. Has to be 'me' or 'I'. Those two are the only first person pronouns... if I recall correctly.

You have several options. I am repeating some of the other answers because it seems practical to have them all together: I hope my plagiarism will be forgiven!

1) Five people went to the party last night, with me being one of them.

This is common and accepted, though not by all purists: it does look a bit informal to me (mostly owing to "with"), which might be fine in the context of a party. "Being" would be a participle; the construction would be called "fused participle", or "accusative with participle" in classicist terms.

2) Five people went to the party last night, with my being one of them.

This is how "me" would traditionally be expressed; but "with" still looks informal, which contrasts weirdly with traditional "my". "Being" would be a gerund here.

3) Five people went to the party last night, I being one of them.

This would be the classical absolute construction. It is impeccably correct, but it sounds rather stiff, especially in this context. "Being" would be a participle.

4) Five people went to the party last night, with myself being one of them.

I know this construction exists, but using "myself" this way might result in some criticism. This would certainly not be my choice.

5) Five people went to the party last night, and I was one of them.

This looks much cleaner. Why use complex constructions in this context if you don't need to? The version with "...; I was one of them" looks OK as well.

I and me are indeed the only first person singular pronouns. Since it is not the subject, me is the correct choice in this instance.

The phrase, while not technially incorrect grammatically, does seem rather awkward when it is not used colloquially. This:

Five people went to the party last night, with me being one of them.

would generally be rephrased.

Five people went to the party last night; I was one of them.

"Me being one of them" is an inversion of the normal subject/object order in order to emphasize the object, me. Since me is still the object, the objective case is still called for.